1. Field of the Invention
The present invention relates to a measurement system, and more particularly, to a measurement system for accurately obtaining an overlay alignment error for a non-measured shot using a high order regression analysis model.
2. Description of the Related Art
Measurement systems used for the characterization of semiconductor devices and the like measure various quantities. For example, a photolithographic process, which is essential for the manufacture of semiconductor devices, repeatedly transfers the same pattern onto a chemical material deposited on a wafer. In general, a pattern region to be exposed by a single exposure is called a shot, and at most 100 or more shots on one paper of 300 mm are exposed. Each of the exposed shots should precisely overlay a shot of a previously formed layer.
A deviation amount of the exposed shot from the shot of the previously formed layer is measured as an overlay alignment error. When the measured overlay alignment error exceeds a desired range, a current process for a lot should be re-started. The overlay alignment error is also used to determine the amount of correction of position necessary for the exposed shot to accurately overlay the shot of the previously formed layer in a process for a subsequent lot.
In a photolithographic process used to produce semiconductor devices, instead of measuring overlay alignment errors for all shots, overlay alignment errors for some shots of a wafer selected in a lot are measured to improve production efficiency and then overlay alignment errors for all shots in the wafer are estimated using the measured overlay alignment errors for the some shots.
A regression analysis method is typically used to measure an overlay alignment error. The regression analysis method may use a regression analysis model that expresses the overlay alignment error in a function of the position of each shot in a wafer. Exposure equipment manufacturers set a specific regression analysis model, and use a regression coefficient of the regression analysis model as an equipment input value for the purpose of alignment correction because the regression coefficient helps to calculate the amount of correction of position necessary for each shot in a wafer to be correctly positioned. Accordingly, calculating a regression coefficient that can accurately provide an overlay alignment error after a photolithographic process is essential for the photolithographic process.
Conventional equipment systems for measuring an overlay alignment error set in advance the number and positions of shots to be measured according to semiconductor manufacturing processes. Accordingly, the conventional equipment systems for measuring the overlay alignment error measure the pertinent shots and output regression coefficients analyzed by a regression analysis model that uses raw data, or is already pre-determined. Accordingly, the conventional equipment systems for measuring the overlay alignment error are very difficult to freely vary the number or positions of shots to be measured in a production line.
Also, most semiconductor manufacturers use a first order regression analysis model to correct an overlay alignment error. However, in the case of the first order regression analysis model, the number or positions of shots to be measured do not greatly affect the reliability of analysis results. Therefore, relatively few attempts have been made to optimize the number or positions of shots to be measured. Some research suggests using experimental methods for repeatedly measuring an overlay alignment error with various combinations of shot numbers and positions, analyzing the measured values, and experimentally determining optimal number and positions of shots to be measured. However, it is difficult to use such an experimental method in a variety of processes and conditions.
As semiconductor devices continue to shrink in size, an overlay alignment error to be measured becomes smaller. Accordingly, analysis results obtained by a first order regression analysis model can no longer satisfy accuracy sufficient for correction of an overlay alignment error. Exposure equipment producers have developed exposure equipment that can form a circuit with a fine line-width by upgrading a light source and hardware, and can be provided with an accurate correction function using a high order regression analysis model.
In the case of the high order regression analysis model, the number and positions of shots to be measured greatly affect the reliability of analysis. Since the high order regression analysis model should measure a greater number of shots than a first order regression analysis method, the positions of shots to be measured need to be optimized in order to achieve highest analysis reliability with least shots. When the positions of shots to be measured are optimized, maximum analysis reliability can be achieved even when the same number of shots are measured.
Furthermore, the reliability of analysis varies depending on the condition of a semiconductor manufacturing process. That is, when the reliability of analysis necessary for accurate correction of an overlay alignment error is fixed and a semiconductor manufacturing process is stably performed, the number of shots to be measured necessary to obtain the reliability of analysis is relatively small, but when the reliability of analysis is fixed and a semiconductor manufacturing process is unstably performed, the number of shots to be measured to obtain the reliability of analysis is relatively large.
Moreover, in order to stably operate a system that can measure a smallest possible number of shots and can accurately correct an overlay alignment error, erroneous data should be detected and filtered. As the order of a regression analysis model increases, the effect of each piece of measured data on regression analysis increases. The effect of each measured data on the regression analysis increases further when the number of measured shots decreases. Accordingly, when the erroneous data is directly used in regression analysis without being filtered, analysis results are greatly distorted. The distortion drastically reduces the accuracy of the correction of overlay alignment errors for subsequent lots. The erroneous data may be caused when overlay marks are damaged due to particles on a wafer during operation or due to other processing. Accordingly, effectively filtering the erroneous data is essential for stably operating the system for accurately correcting the overlay alignment error.
Erroneous data is detected by observing the absolute size of data. That is, after regression analysis, when data has a size or a residual, which is unusually larger than that of other data, the data is detected as erroneous data.
Once erroneous data is detected using a least squares method, regression analysis is performed again on data other than the erroneous data. However, it is possible that a method for detecting erroneous data using a least squares method fails to detect the erroneous data, or mistakes normal data for the erroneous data and removes the normal data. The normal data mistaken for the erroneous data may distort estimates of overlay alignment errors for shots around measured shots. Accordingly, accurate data cannot be obtained using only the conventional method for detecting erroneous data.